Prove that that $U(n)$ is an abelian group.
It’s true that you know that multiplication in $\Bbb Z$ is associative and commutative, but you still have to prove that multiplication in $U(n)$ is associative and commutative, i.e., that multiplication modulo $n$ is associative and commutative. To show that every element of $U(n)$ has a multiplicative inverse in $U(n)$, use Bézout’s lemma: if $a$ and $n$ are relatively prime, there are integers $u$ and $v$ such that $au+vn=1$.